Exact and extremal values of chromatic villainy

Abstract

Chromatic villainy measures the largest number of vertices that must change color to repair a rearrangement of an optimal coloring while preserving every color multiplicity. Weak villainy allows different multiplicities in the repair. On the odd cycle $C_{2r+1}$, we determine the largest repair distance from arbitrary color assignments with a prescribed color-count vector (profile) $\mathbf m$ to proper colorings with any realizable target profile $\mathbf p$: it equals $2r+1-\sum_X\min\{p_X,\lceil m_X/2\rceil\}$. In particular, every feasible profile $(a,b,c)$ has ordinary and weak repair radius $\lfloor a/2\rfloor+\lfloor b/2\rfloor+\lfloor c/2\rfloor$, and $B(C_{2r+1})=r$ for $r\ge2$, answering a question of Clark et al. For arbitrary complete multipartite graphs, we give an arithmetic formula for ordinary villainy with at most $k^2$ candidates and an exact finite formula for weak villainy, where $k$ is the number of parts. Both values are fixed-parameter tractable in $k$ for binary-encoded part sizes. We also determine the maximum of each parameter among all $n$-vertex $k$-chromatic graphs. For $k\ge3$, the ordinary maximum is $\min\{n,4(n-k)\}$, except when $n=4(n-k)-1$ or $(n,k)\in\{(5,3),(7,3)\}$, when it is $n-1$. For $k\ge2$, the weak maximum is $n-\max\{\lceil n/k\rceil,2k-n\}-\varepsilon$, where $\varepsilon\in\{0,1\}$ is determined explicitly by $n$ and $k$. Both extrema are attained by complete multipartite graphs, and a weak extremizer needs at most three distinct part sizes.